No Third Slot: A Local Entropy–Phase Closure of "Quantum Gravity" Within Standard Quantum Mechanics

Within orthodox quantum mechanics, the only empirical content of any physical claim is a change in the outcome statistics of a declared measurement. In the modern formulation of measurement theory this declaration is an instrument (a family of completely positive maps) or equivalently a POVM together with an implementation. In curved spacetimes, localized measurement theory supplies the appropriate notion of a “local interface”: a finite laboratory worldtube with finite switching, whose induced observables lie in the causal hull and whose instruments compose causally. This paper proves a closure result: for any declared localized measurement, all geometric dependence (metric, connection, curvature, horizons) can affect outcome statistics only through exactly two operational apertures: • Phase / support aperture: the implemented localized instrument or channel on the accessible algebra, including any differential proper‑time or propagation‑dependent phase and any geometry‑dependent support of the induced observable. • Entropy aperture: the accessibility restriction itself (restriction of the global state to the accessible algebra), equivalently a trace over inaccessible degrees of freedom, with the corresponding mixedness/entropy as the interface invariant. There is no third independent slot in the quantum‑mechanical probability rule into which “quantum gravity” can be inserted. Common‑mode phases are operationally null. Sub‑resolution “micro‑selection” is inadmissible without a refined POVM/instrument. And “mediator” talk without a mediator POVM is non‑identifiable, because Stinespring dilations are never unique. The closure is robust: within finite‑response classes, small perturbations of couplings or geometry produce proportionally small changes in outcome statistics. The result does not deny the possibility of new physics beyond orthodox QM. It eliminates a specific move that has repeatedly appeared in quantum‑gravity discourse: treating curvature as an additional, independently specifiable ingredient of orthodox quantum‑mechanical prediction at a local interface.